Sebastian van Strien

Numerical calculation of topological entropy for one-dimensional maps. Topological entropy is a measure of the dynamical complexity of a dynamical systems. There are quite a few algorithms for computing topological entropy, but most of them are not very effective. In the case of dynamical systems in dimension one, the situation is better, but there is an obvious gap in the literature. This project aims to fill this gap, and ie expected to result in a publication in an international journal (jointly with supervisor). About one third of the project will consist of reading the literature, and then to implement some algorithms in computer code.
Game dynamics. During the last decades there has been a constant interest in the dynamics of games, in particular in so called replicator dynamics but also in some other classes of dynamical systems associated to games. This project will explore connections between different types of game dynamics and investigate some research questions. The first part of this project will involve reading some recent papers in depth. Depending on the strengths and interests of the student, the project then will continue a more mathematical and numerical approach.
Renormalisation The aim of this project is to go through a recent proof of the Feigenbaum-Coullet-Tresser renormalisation conjectures, of universality in period doubling. These results belong to some of the most remarkable that were obtained in the field of dynamical systems. Current proofs rely on hyperbolic geometry, Teichmuller theory and so-called complex bounds. The project will consist of learning the background material for these proofs, and going through a recent Annals of Mathematics paper by Avila and Lyubich. This project will be an excellent preparation for doing a PhD in dynamical systems and is more on the pure side.
Other projects on low-dimensional dynamics or game dynamics, suggested by students are also most welcome.

Summer Projects (examples)

Dynamics in Games, by Lyndsey Clark (2011-2012).